### Abstract

In the trajectory segmentation problem, we are given a polygonal trajectory with n vertices that we have to subdivide into a minimum number of disjoint segments (subtrajectories) that all satisfy a given criterion. The problem is known to be solvable efficiently for monotone criteria: criteria with the property that if they hold on a certain segment, they also hold on every subsegment of that segment. To the best of our knowledge, no theoretical results are known for nonmonotone criteria. We present a broader study of the segmentation problem, and suggest a general framework for solving it, based on the start-stop diagram: a 2-dimensional diagram that represents all valid and invalid segments of a given trajectory. This yields two subproblems: (1) computing the start-stop diagram, and (2) finding the optimal segmentation for a given diagram. We show that (2) is NP-hard in general. However, we identify properties of the start-stop diagram that make the problem tractable and give a polynomial-time algorithm for this case. We study two concrete nonmonotone criteria that arise in practical applications in more detail. Both are based on a given univariate attribute function f over the domain of the trajectory. We say a segment satisfies an outlier-tolerant criterion if the value of f lies within a certain range for at least a given percentage of the length of the segment. We say a segment satisfies a standard deviation criterion if the standard deviation of f over the length of the segment lies below a given threshold. We show that both criteria satisfy the properties that make the segmentation problem tractable. In particular, we compute an optimal segmentation of a trajectory based on the outlier-tolerant criterion in O(n^{2} log n + kn^{2}) time and on the standard deviation criterion in O(kn^{2}) time, where n is the number of vertices of the input trajectory and k is the number of segments in an optimal solution.

Original language | English (US) |
---|---|

Article number | 26 |

Journal | ACM Transactions on Algorithms |

Volume | 12 |

Issue number | 2 |

DOIs | |

State | Published - Dec 1 2015 |

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### Keywords

- Dynamic programming
- Geometric algorithms
- Segmentation
- Trajectory

### ASJC Scopus subject areas

- Mathematics (miscellaneous)

### Cite this

*ACM Transactions on Algorithms*,

*12*(2), [26]. https://doi.org/10.1145/2660772

**Segmentation of trajectories on nonmonotone criteria.** / Aronov, Boris; Driemel, Anne; Van Kreveld, Marc; Löffler, Maarten; Staals, Frank.

Research output: Contribution to journal › Article

*ACM Transactions on Algorithms*, vol. 12, no. 2, 26. https://doi.org/10.1145/2660772

}

TY - JOUR

T1 - Segmentation of trajectories on nonmonotone criteria

AU - Aronov, Boris

AU - Driemel, Anne

AU - Van Kreveld, Marc

AU - Löffler, Maarten

AU - Staals, Frank

PY - 2015/12/1

Y1 - 2015/12/1

N2 - In the trajectory segmentation problem, we are given a polygonal trajectory with n vertices that we have to subdivide into a minimum number of disjoint segments (subtrajectories) that all satisfy a given criterion. The problem is known to be solvable efficiently for monotone criteria: criteria with the property that if they hold on a certain segment, they also hold on every subsegment of that segment. To the best of our knowledge, no theoretical results are known for nonmonotone criteria. We present a broader study of the segmentation problem, and suggest a general framework for solving it, based on the start-stop diagram: a 2-dimensional diagram that represents all valid and invalid segments of a given trajectory. This yields two subproblems: (1) computing the start-stop diagram, and (2) finding the optimal segmentation for a given diagram. We show that (2) is NP-hard in general. However, we identify properties of the start-stop diagram that make the problem tractable and give a polynomial-time algorithm for this case. We study two concrete nonmonotone criteria that arise in practical applications in more detail. Both are based on a given univariate attribute function f over the domain of the trajectory. We say a segment satisfies an outlier-tolerant criterion if the value of f lies within a certain range for at least a given percentage of the length of the segment. We say a segment satisfies a standard deviation criterion if the standard deviation of f over the length of the segment lies below a given threshold. We show that both criteria satisfy the properties that make the segmentation problem tractable. In particular, we compute an optimal segmentation of a trajectory based on the outlier-tolerant criterion in O(n2 log n + kn2) time and on the standard deviation criterion in O(kn2) time, where n is the number of vertices of the input trajectory and k is the number of segments in an optimal solution.

AB - In the trajectory segmentation problem, we are given a polygonal trajectory with n vertices that we have to subdivide into a minimum number of disjoint segments (subtrajectories) that all satisfy a given criterion. The problem is known to be solvable efficiently for monotone criteria: criteria with the property that if they hold on a certain segment, they also hold on every subsegment of that segment. To the best of our knowledge, no theoretical results are known for nonmonotone criteria. We present a broader study of the segmentation problem, and suggest a general framework for solving it, based on the start-stop diagram: a 2-dimensional diagram that represents all valid and invalid segments of a given trajectory. This yields two subproblems: (1) computing the start-stop diagram, and (2) finding the optimal segmentation for a given diagram. We show that (2) is NP-hard in general. However, we identify properties of the start-stop diagram that make the problem tractable and give a polynomial-time algorithm for this case. We study two concrete nonmonotone criteria that arise in practical applications in more detail. Both are based on a given univariate attribute function f over the domain of the trajectory. We say a segment satisfies an outlier-tolerant criterion if the value of f lies within a certain range for at least a given percentage of the length of the segment. We say a segment satisfies a standard deviation criterion if the standard deviation of f over the length of the segment lies below a given threshold. We show that both criteria satisfy the properties that make the segmentation problem tractable. In particular, we compute an optimal segmentation of a trajectory based on the outlier-tolerant criterion in O(n2 log n + kn2) time and on the standard deviation criterion in O(kn2) time, where n is the number of vertices of the input trajectory and k is the number of segments in an optimal solution.

KW - Dynamic programming

KW - Geometric algorithms

KW - Segmentation

KW - Trajectory

UR - http://www.scopus.com/inward/record.url?scp=84954314158&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84954314158&partnerID=8YFLogxK

U2 - 10.1145/2660772

DO - 10.1145/2660772

M3 - Article

AN - SCOPUS:84954314158

VL - 12

JO - ACM Transactions on Algorithms

JF - ACM Transactions on Algorithms

SN - 1549-6325

IS - 2

M1 - 26

ER -